function M=initializeG(n, k, tol)
% M = RANDORTHMAT(n)
% generates a random n x n orthogonal real matrix.
%
% M = RANDORTHMAT(n,tol)
% explicitly specifies a thresh value that measures linear dependence
% of a newly formed column with the existing columns. Defaults to 1e-6.
%
% In this version the generated matrix distribution *is* uniform over the manifold
% O(n) w.r.t. the induced R^(n^2) Lebesgue measure, at a slight computational 
% overhead (randn + normalization, as opposed to rand ). 
% 
% (c) Ofek Shilon , 2006.


if nargin < 3
    tol=1e-6;
end

M = zeros(n, k); % prealloc

% gram-schmidt on random column vectors

vi = randn(n,1);
% the n-dimensional normal distribution has spherical symmetry, which implies
% that after normalization the drawn vectors would be uniformly distributed on the
% n-dimensional unit sphere.

M(:,1) = vi ./ norm(vi);

for i=2:k
    nrm = 0;
    while nrm<tol
        vi = randn(n,1);
        vi = vi -  M(:,1:i-1)  * ( M(:,1:i-1).' * vi )  ;
        nrm = norm(vi);
    end
    M(:,i) = vi ./ nrm;
end